Energy functional for Lagrangian tori in $\mathbb{C}P^2$

0.17 MB·

by  Hui Ma

ENERGY FUNCTIONAL FOR LAGRANGIAN TORI IN CP2
7 HUIMA,ANDREYE.MIRONOV,ANDDAFENGZUO
1
0
2 Abstract. InthispaperwestudyLagrangiantoriinCP2. Atwo-dimensional
periodic Schro¨dinger operator is associated with every Lagrangian torus in
n
CP2. Weintroduceanenergyfunctionalfortoriasanintegralofthepotential
a
J of the Schro¨dinger operators, which has a natural geometrical meaning (see
below). We study the energy functional on two families of Lagrangian tori
5
and propose a conjecture that the minimum of the functional is achieved by
2
the Clifford torus. We also study deformations of minimal Lagrangian tori.

In particular we show that if the deformation preserves a conformal type of
]
G the torus, then it also preserves the area of the torus. Thus it follows that
deformations generated by Novikov–Veselov equations preserve the area of
D minimalLagrangiantori.
.
h
t
a
m 1. Introduction and main results
[
In this paper we study Lagrangian tori in CP2. The paper consists of two

1 parts. InthefirstpartweintroduceanenergyfunctionalofLagrangiantorusasan
v integralofthepotentialofSchr¨odingeroperatorassociatedwiththetorus. Westudy
1
this energy functional on a family of homogeneous tori and Hamiltonian-minimal
1
Lagrangian tori constructed in [1], respectively. We propose a conjecture that the
2
7 minimum of this functional onthe whole set ofLagrangiantori canbe achievedby
0 the Clifford torus. Similar conjecture was proposed by Montiel and Urbano [2] for
1. their Willmore functional. In the second part of the paper we study deformations
0 ofminimal Lagrangiantori. Such deformationsare relatedto eigenfunctions of the
7 Laplace–Beltramioperatorwitheigenvalueλ=6. UsingNovikov–Veselovhierarchy
1
[3] we propose a method of finding such eigenfunctions. We also prove that if the
:
v deformation of minimal Lagrangian tori preserves the conformal type of the torus
i then it preserves the area of the torus.
X
Let Σ be a closed Lagrangian surface immersed in CP2. Let x,y denote local
r
a conformal coordinates such that the induce metric of Σ is given by
(1.1) ds2 =2ev(x,y)(dx2+dy2).
Let r : U S5 be a local horizontal lift of the immersion defined on an open
subset U of→Σ, where S5 C3 is the unit sphere. Since Σ is Lagrangian and x,y
⊂
are conformal coordinates we have
(1.2) r,r =1, r ,r = r ,r = r ,r =0,
x y x y
h i h i h i h i
Date:January26,2017.
Key words and phrases. Lagrangiansurfaces,Energyfunctional,Novikov–Veselov hierarchy.
ThefirstauthorwassupportedinpartbyNSFCGrantNo.11271213.
ThesecondauthorwassupportedbytheRussianFoundationforBasicResearch(grant16-51-
55012)andbyagrantfromDmitriZimin’sDynastyfoundation.
The third author was supported in part by NSFC (Grant No. 11671371,11371338) and the
Fundamental ResearchFundsfortheCentralUniversities.
1
2 HUIMA,ANDREYE.MIRONOV,ANDDAFENGZUO
(1.3) r ,r = r ,r =2ev,
x x y y
h i h i
and hence
r
R˜ = rx U(3),
|rryx|∈
where , denotes the Hermitianinner|rpy|roductofC3. One candefine a localfunc-
h i
tionβ :U R,calledtheLagrangianangleofΣ,byeiβ(x,y) =detR˜. Consequently
→
r
R=√12e−v2−iβ2rx∈SU(3).
√12e−v2−iβ2ry
 
By direct calculations, by
(1.4) R =AR, R =BR,
x y
where A,B su(3) (see Section 3) we obtain the following lemma.
∈
Lemma 1. [4] The vector function r satisfies the Schro¨dinger equation Lr = 0,
where
iβ iβ
L=(∂ )2+(∂ )2+V(x,y),
x y
− 2 − 2
with the potential
1 i
V =4ev+ (β2+β2)+ β.
4 x y 2△
Let us assume that Σ is a torus given by the mapping
r :R2 S5,
→
where r satisfies (1.2) and (1.3). Then the potential V is a doublely periodic
function with respect to a lattice of periods Λ R2 and r is a Bloch eigenfunction
⊂
of the Schr¨odinger operator L, i.e.,
r((x,y)+λ )=eipsr(x,y), p R, s=1,2,
s s
∈
where λ ,λ is a basis of Λ. Let us introduce the energy of a Lagrangian torus
1 2
as an integral of V (Remark that similarly one can define the energy functional of
arbitrary Lagrangiansurfaces in CP2).
Definition 1. The energy of the Lagrangian torus Σ CP2 is
⊂
1
E(Σ)= Vdx dy.
2 ∧
ZΣ
Remark 1. For convenience, we take the factor 1 in the above definition.
2
It turns out that the energy functional has the following geometric meaning.
Lemma 2. The energy of the Lagrangian torus is
1
E(Σ)= dσ+ H 2dσ,
8 | |
ZΣ ZΣ
where dσ =2evdx dy is the area element of Σ, H is the mean curvature vector.
∧
ENERGY FUNCTIONAL FOR LAGRANGIAN TORI IN CP2 3
Let Σ be a homogeneous torus in CP2, where r ,r ,r are positive numbers
r1,r2,r3 1 2 3
such that r2+r2+r2 =1. Any homogeneous torus can be obtained as the image
1 2 3
of the Hopf projection :S5 CP2 of the 3-torus
H →
(r eiϕ1,r eiϕ2,r eiϕ3),ϕ R S5.
1 2 3 j
{ ∈ }⊂
EveryHomogeneoustorusisHamiltonian-minimalLagrangianinCP2,i.e. its area
is a critical point under Hamiltonian deformations (see [5]). Among homogeneous
tori there is a minimal torus Σ , which is called Clifford torus (we denote
1 , 1 , 1
√3 √3 √3
it by Σ ).
Cl
Proposition 1. The identity
π2(1 r2)(1 r2)(1 r2)
E(Σ )= − 1 − 2 − 3
r1,r2,r3 2r r r
1 2 3
holds. Among Lagrangian homogeneous tori in CP2, the Clifford torus attains the
minimum of the energy functional
4π2
E(Σ )=Area(Σ )= .
Cl Cl
3√3
Let us consider another family of Hamiltonian-minimal Lagrangiantori in CP2
(see [1]). Let Σ CP2 denote a surface given as the image of the surface
m,n,k
⊂
u e2πimy,u e2πiny,u e2πiky S5
1 2 3
{ }⊂
under the Hopf projection, where (u ,u ,u ) R3 such that
1 2 3
∈
u2+u2+u2 =1,
1 2 3
mu2+nu2+ku2 =0,
1 2 3
with integers m n > 0 and k < 0. Such surface Σ is an (immersed or
m,n,k
≥
embedded) Hamiltonian-minimal Lagrangian torus or Klein bottle. The topology
of Σ depends on whether the involution
m,n,k
(u ,u ,u ) (u cos(mπ),u cos(nπ),u cos(kπ))
1 2 3 1 2 3
→
preserves an orientation of the surface mu2+nu2+ku2 =0. We have
1 2 3
Proposition 2. The energy of Σ is greater than the energy of the Clifford
m,n,k
torus
E(Σ )>E(Σ ).
m,n,k Cl
Propositions 1 and 2 allow us to propose the following conjecture.
Conjecture. The Clifford torus attains the minimum of the energy functional
among all Lagrangian tori in CP2.
In the second part of the paper we study deformations of minimal Lagrangian
tori in CP2.
4 HUIMA,ANDREYE.MIRONOV,ANDDAFENGZUO
Theorem 1. Let Σ be a closed minimal Lagrangian torus in CP2 and Σ be a
0 t
smooth deformation of Σ preserving the conformal type of Σ such that Σ is still
0 0 t
minimal Lagrangian. Then the area of Σ is preserved
t
Area(Σ )=Area(Σ ).
t 0
ImportantexamplesofsuchdeformationsaredeformationsgeneratedbyNovikov–
Veselov hierarchy. Let Σ CP2 be a minimal Lagrangian torus. On Σ there are
⊂
coordinatesx,ysuchthattheinducedmetrichastheformds2 =2ev(x,y)(dx2+dy2),
where v satisfies the Tziz´eica equation
(1.5) ∆v =4(e 2v ev).
−
−
Let r :R2 S5 be a lift map for Σ.
→
Theorem 2. [4] There is a mapping r˜(t) : R2 S5, t = (t ,t ,...), r˜(0) = r,
2 3
defining a minimal Lagrangian torus Σ CP2→such that Σ = Σ. The map r˜
t 0
⊂
satisfies the equations
Lr˜=∆r˜+4ev˜r˜,
∂ r˜=A r˜,
tn n
where A are operators of order (2n+1) in x,y and ds2 = 2ev˜(x,y,t)(dx2 +dy2)
n
is the induced metric on Σ . The deformation r˜(t) preserves conformal type of the
t
torus and the spectral curve of the Schro¨dinger operator L. The function V˜ = 4ev˜
with v˜(0)=v satisfies the Novikov–Veselov hierarchy
∂L
=[L,A ]+B L.
n n
∂
tn
Thus Theorems 1 and 2 leads to the following corollary.
Corollary 1. Deformations of minimal Lagrangian tori given by Novikov–Veselov
hierarchy (see Theorem 2) preserve the area of tori.
It would be interesting to use Novikov–Veselovequations to construct deforma-
tionspreservingtheenergyofarbitraryLagrangiantorusinCP2. Similarconstruc-
tion is possible for tori in R3 if one use modified Novikov–Veselov equations (see
[6], [7]).
Deformations of minimal Lagrangian tori are related to eigenfunctions of the
Laplace–Beltrami operator with eigenvalue 6. Generally to find explicitly eigen-
functions of the Laplace–Beltrami operator is a hard problem (see [8]). The pre-
ceding theorem gives a method to find such eigenfunctions for Laplace–Beltrami
operator of minimal Lagrangiantori in CP2.
Theorem 3. Let ds2 = 2ev(x,y)(dx2 + dy2) be a metric on a surface Σ and v
satisfies the Tziz´eica equation. Then functions
s =v2 v2+v v ,
1 x− y xx− yy
s =v v +v
2 x y xy
are eigenfunctions of the Laplace–Beltrami operator with eigenvalue 6.
ENERGY FUNCTIONAL FOR LAGRANGIAN TORI IN CP2 5
Other eigenfunctions can be found with the help of explicit calculations related
to Novikov–Veselovequations.
2. Energy functional
In this section we prove Lemma 1, Propositions 1 and 2.
2.1. Proof of Lemma 2. Recall that the mean curvature vector field on La-
grangian surface Σ CP2 can be expressed in terms of Lagrangian angle β (see
⊂
for example [1])
H =Jgradβ,
where J is the standard complex structure on CP2. With respect to the induced
metric ds2 =2ev(dx2+dy2),
e v
H 2 = Jgradβ 2 = gradβ 2 = − (β2+β2).
| | | | | | 2 x y
Hence
1 i
E(Σ) = 2evdx dy+ 2ev H 2dx dy+ βdx dy
∧ 8 | | ∧ 4 △ ∧
ZΣ ZΣ ZΣ
1
= dσ+ H 2dσ.
8 | |
ZΣ ZΣ
2.2. ProofofProposition1. LetΣ CP2beahomogeneoustorus. Then
after choosing of appropriate coordinra1t,er2s,ra3n⊂d taking automorphisms of CP2, the
horizontal lift r :R2 S5 of Σ can be given by
→ r1,r2,r3
r(x,y)=(r e2πix,r e2πi(a1x+b1y),r e2πi(a2x+b2y)),
1 2 3
where r2+r2+r2 =1. It follows from (1.2) and (1.3) that
1 2 3
r2 r r r r
a =a = 1 , b = 1 3 , b = 1 2 .
1 2 −r2+r2 1 r (r2+r2) 2 −r (r2+r2)
2 3 2 2 3 3 2 3
By direct calculations, one can obtain that the lattice of periods Λ for r, is
Λ= Ze +Ze R2, where H◦
1 2
{ }⊂
r r
e =(r2+r2,0), e =(r2, 2 3).
1 2 3 2 3 r
1
Moreover, it follows from r ,r = r ,r = 4π2r12 that the induced metric,
h x xi h y yi r22+r32
the area, the Lagrangian angle and the potential of the Schr¨odinger operator (see
Lemma 1) of Σ are given by
r1,r2,r3
4π2r2
ds2 = 1 (dx2+dy2),
r2+r2
2 3
dσ = 4π2r r r ,
1 2 3
ZΣr1,r2,r3
1 3r2 r (r2 r2)
β = 2π − 1x 2π 1 2 − 3 y,
r2+r2 − r r (r2+r2)
2 3 2 3 2 3
π2(1 r2)(r2+r2)
V = − 2 1 2 .
r2r2
2 3
6 HUIMA,ANDREYE.MIRONOV,ANDDAFENGZUO
Hence the energy of Σ is given by
r1,r2,r3
(r2r4+r2r2 8r2r2r2+9r4r2r2+r2r4)
E(Σ ) = 4π2r r r +π2 1 2 2 3 − 1 2 3 1 2 3 1 3
r1,r2,r3 1 2 3 2r r r (r2+r2)
1 2 3 2 3
π2(1 r2)(1 r2)(1 r2)
= − 1 − 2 − 3 .
2r r r
1 2 3
Thus it is easy to find that the minimum of E(Σ ) is attained on Clifford
r1,r2,r3
torus Σ .
1 , 1 , 1
√3 √3 √3
2.3. ProofofProposition2. Takinganappropriateparametrizationofthecurve
given by
u2+u2+u2 =1,
1 2 3
mu2+nu2+ku2 =0,
1 2 3
where m n>0 and k <0 are constant integers, we obtain
≥
ψ(u,y)=(u e2πimy,u e2πiny,u e2πiky) S5,
1 2 3
⊂
with
k k ncos2x msin2x
u =sinx ,u =cosx ,u = + .
1 k m 2 k n 3 s n k m k
r − r − − −
Thus we have a horizontal lift of a surface Σ in CP2 given by
r(x,y)=(u (x)e2πimy,u (x)e2πiny,u (x)e2πiky).
1 2 3
Now let us consider the torus given as above, denoted it by Σ .
m,n,k
By straightforwardcalculations, we obtain the induced metric on Σ is
m,n,k
ds2 =2ev1(x)dx2+2ev2(x)dy2,
where
k(m+n (m n)cos(2x))
2ev1(x) = − − ,
−2mn k(m+n)+k(m n)cos(2x)
− −
2ev2(x) = 2kπ2(m+n (m n)cos(2x)).
− − −
Now we have
r
R˜ = rx U(3).
|rryx|∈
By the definition of the Lagrangianan|gryle|detR˜ =eiβ, we get
eiβ =ie2πi(m+n+k)y.
The period in x is e =2π and the period in y is 1, where p=(m k,n k) is
1 p − −
the biggestcommonfactorofm andn. Moreover,the Willmore functionalis given
by
W(Σm,n,k) = H 2dσ = H 22ev1+2v2dxdy
ZΣm,n,k| | Z[0,2π]×[0,p1]| |
= (ev2−2v1βx2+ev1−2v2βy2)dxdy.
ZΣm,n,k
ENERGY FUNCTIONAL FOR LAGRANGIAN TORI IN CP2 7
Thus we have
1 2π √2kπ(m+n (m n)cos2x)
A(Σ ) = dσ = − − − dx,
m,n,k
p 2mn k(m+n)+k(m n)cos2x
ZΣm,n,k Z0 − −
1 2π 2√2p(k+m+n)2π
W(Σ ) = dx,
m,n,k
p 2mn k(m+n)+k(m n)cos2x
Z0 − −
1
E(Σ ) = A(Σ p)+ W(Σ )
m,n,k m,n,k m,n,k
8
1 π 2π 4k(m n)cos2x+( k+m+n)2
= − − dx.
p2√2 2mn k(m+n)+k(m n)cos2x
Z0 − −
Remembering k <0, we have p
4k(m n)cos2x+( k+m+n)2 4k(m n)cos2x+( k+m+n)2
− − − − .
2mn k(m+n)+k(m n)cos2x ≥ 2mn k(m+n) k(m n)
− − − − −
Henpce p
1 π 2π 4k(m n)cos2x+( k+m+n)2
E(Σ ) − −
m,n,k
≥ p2√2 2mn k(m+n) k(m n)
Z0 − − −
π2( k+m+n)2
p
= −
2p m( k+n)
−
π2 m+pr
(pk+n+m)= π2
≥ p − p
m 4
= ( +r)π2 > π2,
p 3√3
where we use 4 0.8, n k =pr and r is a positive integer. Thus the proof of
3√3 ∼ −
Proposition 1.7 is completed.
3. deformations of Minimal Lagrangian tori
Now consider a Lagrangiantorus defined by the composition of the maps
r :R2 S5
→
and . Then R satisfies (1.4) (see [4]), where
H
0 √2ev+2iβ 0
A=−√20ev−2iβ vy +i(ieFveG−v+ 1β ) −v2y +i(ieF−veGv+ 21βy)∈su(3),
2 − 2 y − −
 
0 0 √2ev+2iβ
B = 0 iGe v 1v +i( e vF + 1β ) su(3),
 − − 2 x − − 2 x ∈
−√2ev−2iβ −21vx+i(−e−vF + 21βx) −iGe−v
 
with real functions F and G given by
1
F = ( r ,r ev(v +iβ ))
xy y x x
−2i h i−
and
1
G= ( r ,r ev(v +iβ )).
xy x y y
2i h i−
8 HUIMA,ANDREYE.MIRONOV,ANDDAFENGZUO
The compatibility condition
A B +[A,B]=0
y x
−
leads to the following equations (see [4] and also [9], [10])
2F +2G =(β β )ev,
x y xx yy
−
2F 2G =(β v +β v )ev,
y x x y y x
−
v =4(F2+G2)e 2v 4ev 2(Fβ Gβ )e v.
− x y −
△ − − −
If the torus is minimal then β = const and from the equations it follows that F
and G are constants. After appropriate change of coordinates (a homothety and
rotation) one can assume that F = 1 and G = 0. Hence v satisfies the Tziz´eica
equation (1.5). Smooth periodic solutions of the Tziz´eica equation are finite-gap
solutions. Thesesolutionswerefoundin[11]. MinimalLagrangiantoriwerestudied
in [12]–[17]. Assume that we have a deformation Σ of Σ, Σ = Σ, given by the
t 0
mapping
r(t):R2 S5
→
with the induced metric ds2 = 2ev(x,y,t)(dx2 +dy2), v(x,y,t) satisfies (1.5). We
have R =TR, where
t
is a +ib a +ib
1 1 2 2
(3.1) T = a +ib is a +ib su(3),
1 1 1 3 3
− ∈
a +ib a +ib i(s+s )
2 2 3 3 1
− − −
 
with functions s,s ,a ,a ,a ,b ,b and b depending on x,y and t. From the
1 1 2 3 1 2 3
compatibility conditions
A T +[A,T]=0, B T +[B,T]=0,
t x t y
− −
we obtain the identities
b = e−v2sx, b = e−v2sy,
1 2
2√2 2√2
a = e−2v(√2sy −2e23v(a1vy−2a2x)),
3
4√2
b = e−23v(−8a2−√2e12v(sxvy+syvx−2sxy)),
3
8√2
s = e−23v(8a1+√2e12v(8evs+syvy−sxvx+2sxx))
1
8√2
andthe following overdeterminedsystemof equations for s,a ,a whichdetermine
1 2
the deformation:
1
(3.2) vt+ 2e−2v(−√2e32va2vy−2√2e23va1x+sx)=0,
(3.3) s+12evs=0,
△
(3.4) 2a1x 2a2y a1vx+a2vy √2e−32vsx =0,
− − −
(3.5) 2a1y+2a2x a1vy a2vx+√2e−23vsy =0,
− −
(3.6) √2e32va2vy +2√2e32va1x sx+2e3v(b3vy+sx+s1x)=0,
−
ENERGY FUNCTIONAL FOR LAGRANGIAN TORI IN CP2 9
(3.7) 2e va s +b v =0.
− 3 1y 3 x
−
Now we cangive a geometric interpretationof s. Let r be the normalcomponent
t⊥
of the velocity vector r . Thus
t
b b e v s
r = 1 ir + 2 ir = − (s ir +s ir )=grad .
t⊥ √2ev2 x √2ev2 y 4 x x y y 2
Thus it follows that the deformationwe obtained above is a Hamiltonian deforma-
tionwith Hamiltonian s (see [5]). Moreover(3.3) canbe rewrittenin the following
2
form
s=6s,
△LB
where istheLaplace–Beltramioperatorofthemetric(1.1). Thusthefunction
△LB
s is an eigenfunction of the Laplace–Beltrami operator with the eigenvalue 6.
Proof of Theorem 1. Using (3.2) and (3.4), we get
v = e−v2(2a2y+a2vy +2a1x+a1vx),
t
2√2
s = e23v(2a2y −a2vy−2a1x+a1vx).
x
− √2
Considering the area form dσ =2evdx dy, we set
∧
Ω=∂ (2ev)dx dy = ev2(2a2y+a2vy +2a1x+a1vx)dx dy.
t
∧ √2 ∧
It turns out that Ω=dω, where
ω =√2ev2(a1dy a2dx).
−
If Σ is a smooth deformation of Σ preserving the conformal type of Σ , then
t 0 0
d
2evdx dy = Ω= dω =0,
dt ∧
ZΛ ZΛ ZΛ
where Λ is a lattice of periods. The proof is completed.
(cid:3)
3.1. Novikov–Veselov hierarchy and deformations of minimalLagrangian
tori. In this subsection we consider an example of deformation of minimal La-
grangian tori defined by the second Novikov–Veselov equation. In particular we
give an explicit solution of the system (3.2)–(3.7) in terms of the function v defin-
ing the induced metric (1.1) of the torus.
Let us recall the Novikov–Veselovhierarchy. Let L be a Schr¨odinger operator
L=∂ ∂ +V(z,z¯),
z z¯
and
A =∂2n+1+u ∂2n 1+ +u ∂ +∂2n+1+w ∂2n 1+ +w ∂ ,
2n+1 z 2n−1 z − ··· 1 z z¯ 2n−1 z¯ − ··· 1 z¯
where u =u (z,z¯),w =w (z,z¯). The operator ∂ A defines an evolution
j j j j tn − 2n+1
equation for the eigenfunction r of the Schr¨odinger operator Lr=0 by
∂ r =A r.
tn 2n+1
The Novikov–Veselovequations are
∂L
=[A ,L]+B L,
∂t 2n+1 2n−2
10 HUIMA,ANDREYE.MIRONOV,ANDDAFENGZUO
where B is a differential operator of order 2n 2 and V and coefficients of
2n 2
− −
A are the unknowns. In the case of the periodic Schr¨odinger operators the
2n+1
Novikov–Veselovequations preserve the spectral curve of L.
Example 1. Let n=1. Then
L =∂3+u∂ +∂3+w∂ , B =u +w ,
3 z z z¯ z¯ 0 z z¯
where
u =3V , w =3V .
z¯ z z z¯
The first Novikov-Veselovequation has the following form
V =V +V +uV +wV +Vw +Vu .
t zzz z¯z¯z¯ z z¯ z¯ z
Example 2. In the case n=2, we have
L =∂5+u ∂3+u ∂2+u ∂ +∂5v ∂3+v ∂2+v ∂ ,
5 z 3 z 3z z 1 z z¯ 3 z¯ 3z¯ z¯ 1 z¯
B =u ∂2+u ∂ +w ∂2+w ∂ +u +w ,
2 3z z 3zz z 3z¯ z¯ 3z¯z¯ z¯ 1z 1z¯
where u ,w satisfies the equations.
j j
(3.8) u =5V , w =5V ,
3z¯ z 3z¯ z¯
(3.9) u =10V +3u V +u V u ,
1z¯ zzz 3 z 3z − 3zzz¯
(3.10) w =10V +3w V +w V w .
1z z¯z¯z¯ 3 z¯ 3z¯ − 3zz¯z¯
The second Novikov–Veselovequation has the following form
V =∂5V +u V +2u V +(u +u )V +
t z 3 zzz 3z zz 1 3zz z
∂5V +w V +2w V +(w +w )V +(w +u )V.
z¯ 3 z¯z¯z¯ 3z¯ z¯z¯ 1 3z¯z¯ z 1z¯ 1z
It turns out that if we assume that V = ev(z,z¯), where v is a real function
satisfyingtheTziz´eicaequation,thenthefirststationaryNovikov–Veselovequation
follows from the Tziz´eica equation.
Theorem 4. [18] The stationary Novikov–Veselov equation
[A ,L]+B L=0,
3 0
with A =∂3+∂3 (v2+v )∂ (v2+v )∂ , B = ∂ (v2+v ) ∂ (v2+v ),
3 z z¯− z zz z− z¯ z¯z¯ z¯ 0 − z z zz − z¯ z¯ z¯z¯
is equivalent to the following equations
∂ (e 2v ev v )+2v (e 2v ev v )=0,
z − zz¯ z − zz¯
− − − −
∂z¯(e−2v ev vzz¯)+2vz¯(e−2v ev vzz¯)=0,
− − − −
which follows from the Tziz´eica equation.
Moreover,coefficientsofA canbe expressedintermsofv andderivativesof
2n+1
v. In particular it means that the equations (3.8)–(3.10) can be solved explicitly.